DiscreteDists (version 1.1.3)

dPTRTE: The Poisson–transmuted record type exponential distribution

Description

These functions define the density, distribution function, quantile function and random generation for the Poisson–transmuted record type exponential (PTRTE) distribution with parameters \(\mu\) and \(\sigma\).

This distribution was proposed by Erbayram and Akdogan (2025) as a new discrete distribution obtained from a mixed Poisson model, where the Poisson parameter follows a transmuted record type exponential distribution.

Usage

dPTRTE(x, mu, sigma, log = FALSE)
pPTRTE(q, mu, sigma, lower.tail = TRUE, log.p = FALSE)
qPTRTE(p, mu, sigma, lower.tail = TRUE, log.p = FALSE)
rPTRTE(n, mu, sigma)

Value

dPTRTE gives the density, pPTRTE gives the distribution function, qPTRTE gives the quantile function, rPTRTE generates random deviates.

Arguments

x, q

vector of (non-negative integer) quantiles.

mu

vector of the mu parameter.

sigma

vector of the sigma parameter.

log, log.p

logical; if TRUE, probabilities p are given as log(p).

lower.tail

logical; if TRUE (default), probabilities are \(P[X <= x]\), otherwise, \(P[X > x]\).

p

vector of probabilities.

n

number of random values to return.

Author

Rebeca Isabel Rodriguez Gonzalez,

Details

The Poisson–transmuted record type exponential distribution with parameters \(\mu\) and \(\sigma\) has support \(x = 0,1,2,\dots\) and probability mass function given by

$$f(x | \mu, \sigma) = \frac{\mu}{(1+\mu)^{x+1}} \left(\frac{\sigma \mu (1+x)}{1+\mu} - (\sigma-1) \right)$$

with \(\mu > 0\) and \(0 < \sigma < 1\).

References

Erbayram, T., & Akdogan, Y. (2025). A new discrete model generated from mixed Poisson transmuted record type exponential distribution. Ricerca di Matematica, 74, 1225–1247.

See Also

PTRTE.

Examples

# Example 1
# Plotting the mass function for different parameter values

x_max <- 20
probs1 <- dPTRTE(x=0:x_max, mu=0.5, sigma=0.5)
probs2 <- dPTRTE(x=0:x_max, mu=1, sigma=0.5)
probs3 <- dPTRTE(x=0:x_max, mu=2, sigma=0.5)

# To plot the first k values
plot(x=0:x_max, y=probs1, type="o", lwd=2, col="dodgerblue", las=1,
     ylab="P(X=x)", xlab="X", main="Probability for PTRTE",
     ylim=c(0, 0.80))
points(x=0:x_max, y=probs2, type="o", lwd=2, col="tomato")
points(x=0:x_max, y=probs3, type="o", lwd=2, col="green4")
legend("topright", col=c("dodgerblue", "tomato", "green4"), lwd=3,
       legend=c("mu=0.5, sigma=0.5",
                "mu=1, sigma=0.5",
                "mu=2, sigma=0.5"))

# Example 2
# Checking if the cumulative curves converge to 1

x_max <- 10

cumulative_probs1 <- pPTRTE(q=0:x_max, mu=1, sigma=0.5)
cumulative_probs2 <- pPTRTE(q=0:x_max, mu=1, sigma=0.7)
cumulative_probs3 <- pPTRTE(q=0:x_max, mu=1, sigma=0.9)

plot(x=0:x_max, y=cumulative_probs1, col="dodgerblue",
     type="o", las=1, ylim=c(0, 1),
     main="Cumulative for PTRTE",
     xlab="X", ylab="Probability")
points(x=0:x_max, y=cumulative_probs2, type="o", col="tomato")
points(x=0:x_max, y=cumulative_probs3, type="o", col="green4")
legend("bottomright", col=c("dodgerblue", "tomato", "green4"), lwd=3,
       legend=c("mu=1, sigma=0.5",
                "mu=1, sigma=0.7",
                "mu=1, sigma=0.9"))

# Example 3
# Comparing the random generator output with
# the theoretical probabilities

x_max <- 30
probs1 <- dPTRTE(x=0:x_max, mu=0.5, sigma=0.5)
names(probs1) <- 0:x_max

x <- rPTRTE(n=1000, mu=0.5, sigma=0.5)
probs2 <- prop.table(table(x))

cn <- union(names(probs1), names(probs2))
height <- rbind(probs1[cn], probs2[cn])
mp <- barplot(height, beside=TRUE, names.arg=cn,
              col=c("dodgerblue3","firebrick3"), las=1,
              xlab="X", ylab="Proportion")
legend("topright",
       legend=c("Theoretical", "Simulated"),
       bty="n", lwd=3,
       col=c("dodgerblue3","firebrick3"), lty=1)

# Example 4
# Checking the quantile function

mu <- 1
sigma <- 0.5
p <- seq(from=0, to=1, by=0.01)
qxx <- qPTRTE(p=p, mu=mu, sigma=sigma, lower.tail=TRUE, log.p=FALSE)
plot(p, qxx, type="s", lwd=2, col="green3", ylab="quantiles",
     main="Quantiles of DPTRTE(mu=1, sigma=0.5)")